MinT - Best Known (72−7, 72, s)-Nets in Base 4

Best Known (72−7, 72, s)-Nets in Base 4

(72−7, 72, 2811623)-Net over F₄ — Constructive and digital

Digital (65, 72, 2811623)-net over F₄, using

net defined by OOA [i] based on linear OOA(4⁷², 2811623, F₄, 9, 7) (dual of [(2811623, 9), 25304535, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(4⁷², 2811624, F₄, 3, 7) (dual of [(2811624, 3), 8434800, 8]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(4¹¹, 15423, F₄, 3, 3) (dual of [(15423, 3), 46258, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(4¹¹, 15423, F₄, 2, 3) (dual of [(15423, 2), 30835, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(4¹¹, 15423, F₄, 3) (dual of [15423, 15412, 4]-code or 15423-cap in PG(10,4)), using
        caps completed using a computer search [i]
    2. linear OOA(4⁶¹, 2796201, F₄, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
      - OOA 3-folding [i] based on linear OA(4⁶¹, large, F₄, 7) (dual of [large, large−61, 8]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(72−7, 72, large)-Net over F₄ — Digital

Digital (65, 72, large)-net over F₄, using

t-expansion [i] based on digital (64, 72, large)-net over F₄, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁷², large, F₄, 8) (dual of [large, large−72, 9]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]

(72−7, 72, large)-Net in Base 4 — Upper bound on s

There is no (65, 72, large)-net in base 4, because

5 times m-reduction [i] would yield (65, 67, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]